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Journal of Physics B Atomic Molecular and Optical Physics

Analytical quantal collisional Stark mixing probabilities

D. Vrinceanu, M. Raymond Flannery

Published Dec 14, 2000• 23 citations
Publisher

Abstract

An exact expression for the probability P(n)l'l of Stark mixing transitions between arbitrary angular momentum states l and l' within the same energy shell n, as a result of a collision with a slow charged projectile at large impact parameters is presented. The formula obtained is compact and easy to use for numerical evaluations even for very large quantum numbers (n~100). A classical approximation is directly obtained and compared with the exact quantal result in the limit of large n. Two distinct sets of quantal oscillations are predicted.

DOI
10.1088/0953-4075/34/1/101

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Physical Review A•Oct 1, 1996•26 citations
An exact expression for the probability P(n)l'l of Stark mixing transitions between arbitrary angular momentum states l and l' within the same energy shell n, as a result of a collision with a slow charged projectile at large impact parameters is presented. The formula obtained is compact and easy to use for numerical evaluations even for very large quantum numbers (n~100). A classical approximation is directly obtained and compared with the exact quantal result in the limit of large n. Two distinct sets of quantal oscillations are predicted.
Exact classical and quantal solutions are presented for the full array of intrashell transitions $n\mathcal{l}\ensuremath{\rightarrow}n{\mathcal{l}}^{\ensuremath{'}},$ between any angular momentum states, induced by slow distant collisions with a charged particle. The collisions are adiabatic with respect to the orbital frequency of the atomic electron and the transitions are induced by the weak ion-atom dipole field generated by the ion moving along a classical path. The rich symmetry of the problem allows a unified approach and is the source of the excellent agreement, beyond the usual Ehrenfest's correspondence principle, between the classical and quantal treatments. A classical transition probability is defined. Probabilities for transition between any angular momentum states within a high Rydberg energy level are derived in exact analytic forms and are analyzed for a large number of numerical examples. The transition probabilities obtained from the three methods---quantal and classical formulations and Monte Carlo classical simulations---are directly compared to provide excellent agreement.
First Major Highlight: Collisional Stark Mixing at ultralow energies in Rydberg Plasmas. Atoms in high (n,l) states formed in cold Rydberg plasmas decay to the ground state in a succession of radiative transitions populating intermediate excited states. Which states are populated during cascading and with what probability can be obtained from known transition probabilities. Classical models simplify the calculation considerably, revealing the "trajectory" in nt space followed during the cascade, scaling rules and other aspects hidden within the quantal approach. Conserved Quantity (quantal analogue obscure): Quantal-Classical correspondence in radiative recombination and decay is directly demonstrated. Classical transition probabilities and radiative, recombination cross-section are in excellent agreement with quantal results. Cross section for three body collisional capture is formulated for arbitrary electron-energy distributions.
Simple analytical expressions for the cross sections of nl to n'l' transitions from Rydberg-neutral inelastic collisions are obtained on the basis of the free electron model for electron-perturber scattering. The dependences of the electron-perturber scattering amplitude on the electron momentum and scattering angle are allowed for. The analytical results agree well with available numerical calculations. The analysis of the equations obtained yields a conclusion that the cross sections at l<or=l' depend weakly on l and l' for degenerate states and are proportional to the statistical weights of the final states for transitions with large energy defects.
The intercombination transitions between l-resolved Rydberg levels due to collisions with fast electrons are studied within the Ochkur approximation for principal quantum numbers n\ensuremath{\le}20, where l is the angular momentum. The use of the Heisenberg correspondence principle for radial integrals enables one to obtain analytic expressions for any term in the multipole expansion of the cross section for collisions with an arbitrary n and l change. For transitions involving angular momenta that are small compared to the principal quantum numbers, the semiclassical and the exact results are found to be fairly close, within the accuracy consistent with the Born approximation. As a result, the semiclassical approach furnishes a ready estimate for cross sections, and its validity range proves broad enough to cover the region of large momenta transferred to the atom. The numerical results also confirm that dipole transitions have no dominance over the collisions with other \ensuremath{\Delta}l values, in accord with the general model of intercombination scattering.
An approximate scaling formula has been determined that permits the rapid estimation of cross sections for angular-momentum-changing collisions of Rydberg atoms with a variety of targets, using information about low-energy-electron scattering from the target. The formula is obtained by fitting the results of coupled-channel and Born-approximation calculations to functions of reduced parameters. Application to Na(nd)\mathrm{Na}(\mathrm{nd})Na(nd) + He, Ne, Ar, N2{\mathrm{N}}_{2}N2​, CH4{\mathrm{H}}_{4}H4​, C3{\mathrm{C}}_{3}C3​H8{\mathrm{H}}_{8}H8​; Rb(nf)\mathrm{Rb}(\mathrm{nf})Rb(nf) + He, Ar, Xe; and Xe(nf)\mathrm{Xe}(\mathrm{nf})Xe(nf) + CO2{\mathrm{O}}_{2}O2​, suggests that the accuracy is about a factor of 2.
A theory of inelastic n,l-changing and quasielastic l-mixing collisions of Rydberg atoms with the heavy rare-gas atoms is developed. It is based on the semiclassical impact-parameter approach combined with the normalized perturbation theory. Semiclassical formulas for scaled transition probabilities and cross sections as functions of the inelasticity parameter and the collision strength are analyzed. This approach gives a general description of collisional nl\ensuremath{\rightarrow}${\mathit{n}}^{\ensuremath{'}}$ transitions independently of the specific values of the principal quantum number, the relative velocity, and the type of colliding atoms. The energy and angular dependence of the electron-atom scattering amplitude is included in the theory by incorporating the impulse-approximation results in the weak-coupling limit. The Ramsauer-Townsend effect affects significantly the values of the cross sections, especially for the inelastic transitions with large energy transfer. The results obtained are used in calculations of the quenching cross sections for the Rb(nS,nD,nF) atoms in collisions with Ar, Kr, and Xe. Comparison of theory and experiment is made in a wide range of the principal quantum numbers and transition energy defects. \textcopyright{} 1996 The American Physical Society.